Question

1. Determine whether the following set is linearly independent or not. Prove your clas a. [1+1, 2+2-2,1 +32"} b. {2+1, 3x +3',-6 +2"} 8. Let T be a linear transformation from a vector space V to W over R. . Let .. . be linearly independent vectors of V. Prove that if T is one to one, prove that (un)....(...) are linearly independent. (m) is ) be a spanning set of V. Prove that it is onto, then Tu... h at .... a spanning set of W.

Answer 1

no 7 Tag {ita, x+222 , 14 322 } let 2, (HW) + d2 (211 2012) &dz (14 342) 00 for d,, dz, dz be three sceler as dit die +2₂4 +222 w² +23+ 342 12 20 :) (21+43) + x (21+22) + x² (242+363) 20 ditd, 20 -2 1 2d21 322²0 - ② - gives dz dz = 0 Now

1:3x + gines 52₂ 0 gy eq ③ 4 = -dz By en ② dzo cha hence for diid, de be three stalen with ith 2 (1+2)+ L2 (2x+ 2012) +d3 (14 342) 20 => 2120, 4220, 2250 Sitx, n.21?, 14 342) is L'I (6) {214, 36142, - Sta2} considen 4 (2+2) + d2( 344 x 2) +43 (+6142) where d,, &, da be three scolenn 3) 201 - 6dz tal d1+342) & u2( 229 23220 27 221-62320 ) 2,-343 20 - dit) 2₂ 200 de 12, 20③

now ea 6 - 0 giren 32, 13d2 = 0 & det dz 20 v harde which is the ern By er 2 = 343 . This showes that by taking 2321 we get dga3 to, d221 to, 2321 to such that 2, (214) + d( 3x +22) at tgl - 6+22)zo 1 -> 1214, 3x442, -6+ 42) is LD

I use the definition of linearly independent and dependent to solve this problem